Chapter 3 The force with no direction
Genetic drift
In every finite population, which gene copies get passed on is partly luck: it has no direction, yet it erases variation at a rate you can measure.
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Genetic drift is the random change in how common a gene variant is, caused by chance in which individuals survive, mate and pass on which of their two gene copies. It happens in every population of finite size. It has no direction: a variant is as likely to drift up as down. But its effects are not random in the long run. Drift eventually fixes one variant and loses the others, it removes genetic variation at a rate set by population size, and a neutral variant’s chance of taking over is exactly its current frequency. This chapter builds those three results from one picture and then tests them against an experiment run in the 1950s on 107 populations of fruit flies.
The previous chapter’s equations had no chance in them: give selection a value and the allele frequency follows a curve. That works for a population so large that the next generation’s share of A equals its expected share. Real populations are not infinite. A beetle population of 48 passes on 96 gene copies, no more, and which 96 is a lottery.
The next generation is a sample of the last
The standard model of drift was formulated by Sewall Wright and R. A. Fisher around 1930 and is named after both[1][2]. The Wright–Fisher model makes three idealizations: a fixed number N of diploid individuals, so 2N gene copies; generations that do not overlap; and reproduction in which each gene copy of the new generation picks a parent copy from the old one, at random and with replacement. Picking with replacement means some parent copies are picked several times and some not at all, which is what happens when some individuals have many offspring and others none.
Count the orange copies in a new row. Each of the 2N picks is orange with probability p, the share of orange in the row above, independently of the others. So the number of orange copies in the next generation is a binomial random variable[2]:
Everything in this chapter follows from those two moments. The mean says drift has no direction. The variance says how hard it shakes the frequency: the standard deviation of one generation’s change is √(pq/2N). In a population of 20 beetles at p = 0.5 that is 0.08, eight percentage points per generation. In a population of 20,000, it is a quarter of a point. Drift is always there; it is only large when N is small.
Every simulation on this site draws that binomial number exactly, from a seeded random-number generator, rather than approximating it with a normal distribution. That matters near p = 0 and p = 1, where the normal approximation would allow frequencies below zero or above one.
No direction, but no return
Three things are visible in Figure 2, and each is a theorem.
The average does not move. E[p′] = p in equation (3.1), so the expected frequency is the same every generation, forever[2]. Drift is directionless. The populations that drift up balance those that drift down. Biologists sometimes say drift “removes rare alleles”; it would be more accurate to say that a rare allele starts closer to zero, so a directionless walk is more likely to hit zero first.
The populations spread apart. The variance in equation (3.1) adds up generation after generation, so populations that started identical become more and more different. This is the simplest reason why isolated populations of the same species differ genetically even when they live in identical environments, and it is why the idea of a “typical” population is a statistical artefact.
Every population ends up fixed. Once a population has lost orange, or lost blue, nothing in the model can bring it back: 0 and 2N are absorbing states. Without mutation or migration, every population eventually reaches one of them. With N = 20 and p = 0.5 the exact Wright–Fisher chain gives an average of 54 generations to absorption, close to the diffusion approximation 4N ln 2 ≈ 2.8N[3]. Raise N to 200 and the wait is ten times longer, which is why the time scale of drift is measured in units of N generations.
The chance of winning is the current frequency
If drift has no direction, but every population ends fixed for one allele or the other, how often does each allele win? There is a short argument that gives the answer exactly. Go far enough into the future and every gene copy in the population will descend from a single copy alive today, because lineages keep dying out (the hollow circles in Figure 1) until only one is left. Under neutrality every one of today’s 2N copies has the same chance of being that ancestor. So the chance that the eventual winner is an A copy is the fraction of copies that are A[2]:
A new neutral mutation starts as one copy among 2N, so its chance of eventually taking over the population is 1/2N. In a population of a million that is one in two million. Most new mutations are lost within a few generations, by chance alone. The rare one that wins takes a long time: Kimura and Ohta showed that, conditional on fixing, a new neutral mutation takes on average about 4N generations to get there[3]. The site’s exact Markov-chain calculation for N = 50 gives 197 generations, against 4N = 200.
With forty populations the count of winners scatters by a few from run to run; that scatter is a binomial too, with standard deviation √(40 × 0.2 × 0.8) ≈ 2.5. Run 200 populations instead and the fraction fixed settles within a few percent of p₀. The theorem does not depend on N: small populations reach the outcome faster, but the odds are the same.
There is a striking consequence for molecular evolution. If neutral mutations arise at rate μ per gene copy per generation, a population produces 2Nμ of them each generation, and each fixes with probability 1/2N. The rate at which neutral substitutions accumulate is therefore 2Nμ × 1/2N = μ, independent of population size[2]. Large populations produce more mutations, small ones fix each with higher probability, and the two effects cancel exactly. This is why neutral DNA differences between species accumulate roughly like a clock. established
Variation decays at a rate of 1/2N
The lower panel of Figure 2 measured something else: the average heterozygosity, H = 2pq, the chance that two gene copies drawn at random from a population are different alleles. It falls smoothly even though each population jumps about. Here is why. Pick two distinct copies in the new generation. With probability 1/2N they were copied from the same parent copy, and so are identical. Otherwise they come from two different parent copies, which differ with probability H. So
This is exact for the expected value in the Wright–Fisher model[2]; the engine behind this site checks it against the full probability distribution to twelve decimal places. A population loses a fraction 1/2N of its variation every generation, so half of it in about 1.4N generations: 28 generations for 20 beetles, 140 for 100, 1.4 million for a million. That is the population-genetic case for large populations in conservation: heterozygosity, once lost, comes back only through mutation or migration, which is slow.
Drag the population size in this sentence, 20 beetles, and Figure 2 reruns with the new value; the dashed curve in its lower panel is equation (3.3).
Census size and effective size
Equation (3.1) has N in it, but which N? A herd of 100 elephant seals in which one bull fathers every pup does not drift like a Wright–Fisher population of 100. Each pup has one of the herd’s hundred mothers, but all of them share a father, so half of every pup’s genes come through a single individual. The effective population size Ne is defined as the size of an ideal Wright–Fisher population that would lose heterozygosity at the same rate as the real one. For a population with Nm breeding males and Nf breeding females, tracing two gene copies back one generation gives[2]
With one bull and 100 cows, Ne = 400/101 ≈ 4: a herd of 101 animals drifts like a population of four. With 5 bulls and 50 cows it is about 18. Fluctuating size has a similar effect: because heterozygosity is lost multiplicatively, generation by generation, the long-run Ne of a population whose size changes is the harmonic mean of its sizes, which is dominated by the smallest ones[2]. A population of a million that passes through ten individuals once every hundred generations has a long-run Ne under a thousand. And variance in family size, some parents leaving many offspring and others none, lowers Ne further. All three push in the same direction, which is why Ne is nearly always smaller than the head count, often by a factor of ten or more.
Buri’s flies
All of this could be a property of a mathematical model and nothing more. The classic test is an experiment that Peter Buri published in 1956[4]. He used two alleles of the brown eye-colour gene of Drosophila melanogaster, bw and bw75, which are codominant (all three genotypes can be told apart by eye colour) and, as far as he could tell, neutral in his culture conditions. In the series whose counts are shown here (his Series I; a second series used 105 bottles) he set up 107 bottles, each with 16 flies, 8 males and 8 females, all heterozygous: 32 gene copies per bottle, exactly 16 of them bw75, p = 0.5 everywhere. Each generation he picked 8 males and 8 females at random from each bottle’s offspring to found the next bottle, and counted the bw75 copies. He did this for 19 generations. documented
The counts survive[5], and Figure 4 shows them generation by generation: how many of the 107 bottles had 0, 1, 2 … 32 copies of bw75.
The histogram does what the theory says it should. In generation 1 the populations cluster around 16 copies. By generation 7 the distribution is flat: a bottle is as likely to have 5 copies as 25. From then on populations pile up at the two edges, and by generation 19, 30 bottles had lost bw75 completely and 28 had fixed it. The average frequency across all 107 bottles was still 0.47. A directionless process, run 107 times, produced a U. established
But set the model to Ne = 16, the number of flies Buri actually transferred each generation, and it is too slow: it predicts about 12.5 lost and 12.5 fixed at generation 19, well under half of what Buri saw, and its heterozygosity curve sits above every data point after the first few generations. The data drift as if each bottle had about 9 flies; the model with Ne = 9 predicts 28 lost and 28 fixed. Fitting equation (3.3) to the mean heterozygosity of the 107 bottles by least squares gives Ne ≈ 9.4. Some of the 16 flies left many offspring and others few or none, and the variance in reproductive success shrinks the effective size[2]. Buri’s flies are the standard demonstration that the census count is an upper bound on the effective size, not a measure of it.
Drift compared with selection and gene flow
Students are often asked to tell drift apart from the other forces. The cleanest way is by what each one does to the allele frequency, and what it depends on:
| Direction | Strength set by | Effect on populations | |
|---|---|---|---|
| Genetic drift | None (E[Δp] = 0) | Population size: variance pq/2Ne | Makes populations differ; erodes variation within each |
| Natural selection | Toward fitter genotypes | Fitness differences s, and pq | Adapts; can make populations differ or alike |
| Gene flow | Toward the migrants’ frequency | Migration rate m | Makes populations alike; brings in variation |
| Mutation | Toward the mutation balance | Mutation rates, about 10−8 per base | The only source of new variants |
Two misconceptions follow from the table. Drift is not the same as a catastrophe. Founder events and bottlenecks (the next chapter) are periods of unusually strong drift, but drift happens every generation in every finite population, including large, stable ones. And drift is not a weak force that only matters in tiny populations. Its strength per generation is small when Ne is large, but it acts on every gene at once, it never stops, and for any variant whose effect on fitness is small enough, it is the only force that matters.
Why drift matters
Drift was once treated as a nuisance, a source of noise in the “real” business of selection. Three results above changed that. First, because the fixation probability of a new neutral mutation is 1/2N and the substitution rate is μ, a large part of the DNA differences between species can be explained without any selection at all; this is the neutral theory of molecular evolution, developed by Motoo Kimura from the late 1960s[6][2]. Second, because heterozygosity falls at rate 1/2Ne, and Ne is often far below the census count, small and fragmented populations of endangered species lose genetic variation faster than their numbers suggest. Third, drift does not stop when selection starts. A beneficial mutation is still a single copy at first, and a single copy is at the mercy of the lottery in Figure 1. How often luck beats a real advantage is the subject of Chapter 6.
Before that, two special cases of drift that have names of their own: what happens when a population is founded by a handful of individuals, or crashes to a handful and recovers.